probability distribution functions in physicsProbabilityDistributionFunctionsInPhysics2009-04-22 12:45:272009-04-22 13:03:07Topicprobability distribution functionprobability measureassociated probability measurePoisson distributiondpdfdiscrete probability distribution functiondistribution of $X$ on $I$cumulative distribution functioncdfRadon--Nikodym derivativeGaussian distributionnormal distributionFermi-Dirac distribution function%%Planet physics followed by special preamble
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\newcommand{\sequence}[1]{\{#1\}}This is a contributed topic on probability distribution functions and their
applications in physics (mostly in quantum mechanics and statistical mechanics).
\section{Probability Distribution Functions in Physics}
One needs to introduce first a Borel space $\borel$, then a measure space $(\Omega, \borel, \mu)$, and finally define a real function that is measurable `almost everywhere' and is also normalized to unity.
Let $(\Omega, \borel, \mu)$ be a measure space. A {\em probability distribution function} on (the domain) $\Omega$ is a function $f: \Omega \longrightarrow \reals$ such that:
\begin{enumerate}
\item $f$ is $\mu$-measurable
\item $f$ is nonnegative $\mu$-almost everywhere.
\item $f$ satisfies the equation
$$
\int_{\Omega} f(x)\ d\mu = 1
$$
\end{enumerate}
Thus, a probability distribution function $f$ induces a {\em probability measure} $P$ on the measure space $(\Omega, \borel)$, given by
$$
P(X) := \int_X f(x)\ d\mu = \int_{\Omega} 1_X f(x)\ d\mu,
$$
for all $X \in \borel$. The measure $P$ is called the {\em associated probability measure} of $f$. Note that $P$ and $\mu$ are different measures, \PMlinkescapetext{even} though they both share the same underlying measurable space $(\Omega, \borel)$.
\section{Examples}
\subsection{Fermi-Dirac Distribution}
This is a widely used probability distribution function applicable to all fermion particles in quantum statistical mechanics.
\subsection{The Discrete Case: dpdf}
Consider a countable set $I$ with a counting measure imposed on $I$, such that $\mu(A) := |A|$, is the cardinality of $A$, for any subset $A \subset I$. A {\em discrete probability distribution function (\bf dpdf)} $f_d$ on $I$ can be then defined as a nonnegative function $f_d : I \longrightarrow \reals$ satisfying the equation
$$
\sum_{i \in I} f_d(i) = 1.
$$
A simple example is the Poisson distribution $P_r$ on $\naturals$ (for any real number $r$), which is given by
$$
P_r(i) := e^{-r} \frac{r^i}{i!}
$$
for any $i \in \naturals$.
Given any probability (or measure) space $(\Omega, \borel, \mu)$ and any random variable $X: \Omega \longrightarrow I$, one costructs a distribution function on $I$ by taking $f(i) := \mu(\{X = i\})$. The resulting function $\Delta$ is called the {\em distribution of $X$ on $I$}.
\subsection{The Continuous Case: cpdf}
Let $(\Omega, \borel, \mu)$ equals $(\reals, \borel_\lambda, \lambda)$ be the set of real numbers equipped with a Lebesgue measure. Then a continuous probability distribution function ({\em cpdf}) $f_c : \reals \longrightarrow \reals$ is simply a measurable, nonnegative almost everywhere function such that
$$
\int_{-\infty}^\infty f_c(x)\ dx = 1.
$$
The associated measure has a \PMlinkexternal{Radon--Nikodym derivative}{RadonNikodymTheorem} with respect to $\lambda$ equal to $f_c$:
$$
\frac{dP}{d\lambda} = f_c.
$$
One defines the {\em cummulative distribution function, or {\bf cdf},} $F$ of $f_c$ by the formula
$$
F(x) := P(\{X \leq x\}) = \int_{-\infty}^x f(t)\ dt,
$$
for all $x \in \reals$. A well known example of a probability distribution function on $\reals$ is the {\em Gaussian distribution}, or {\em normal distribution}
$$
f(x) := \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-m)^2/2\sigma^2}.
$$